Self number

Discovered in 1949 by the Indian mathematician D.R. Kaprekar a self number or Colombian number is an integer which, in a given base, can not be generated by any other integer added to the sum of its digits. For example, 21 is not a self number, since it can be generated by the sum of 15 and its digits, that is, 21 = 15 + 1 + 5. There is no such sum for 20, hence it is a self number. The first few base 10 self numbers are

1, 3, 5, 7, 9, 20, 31, 42, 53, 64, 75, 86, 97, 108, 110, 121, 132, 143, 154, 165, 176, 187, 198, 209, 211, 222, 233, 244, 255, 266, 277, 288, 299, 310, 312, 323, 334, 345, 356, 367, 378, 389, 400, 411, 413, 424, 435, 446, 457, 468, 479, 490, 501, 512, 514, 525

In general, for even bases, all odd numbers below the base number are self numbers, since any number below such an odd number would have to also be a 1-digit number which when added to its digit would result in an even number. For odd bases, all odd numbers are self numbers.

The following recurrence relation generates base 10 self numbers:

$C_{k}=8\cdot 10^{k-1}+C_{k-1}+8$

(with C₁ = 9)

And for binary numbers:

$C_{k}=2^{j}+C_{k-1}+1$

(where j stands for the number of digits) we can generalize a recurrence relation to generate self numbers in any base b:

$C_{k}=(b-2)b^{k-1}+C_{k-1}+(b-2)$

in which C₁ = b - 1 for even bases and C₁ = b - 2 for odd bases.

The existence of these recurrence relations shows that for any base there are infinitely many self numbers.